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Lecture 2

Lecture 2. Method of Functional Separation of Variables. Andrei D. Polyanin. Functional Separation of Variables. General form of exact solutions:. In general, the functions j m ( t ), y m ( t ), and F ( z ) in ( * ) are not known in advance and are to be identified.

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Lecture 2

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  1. Lecture 2 Method of FunctionalSeparation of Variables Andrei D. Polyanin

  2. Functional Separation of Variables General form of exact solutions: In general, the functions jm(t), ym(t), and F(z) in (*) are not known in advance and are to be identified. Main idea: the functional-differential equation resulting from the substitution of expression (*) in the original PDE should be reduced to the standard bilinear functional equation (Lecture 1: Method of generalized separation of variables). Functional separable solutions of special form: The former solution is called a generalized travelling-wave solution.

  3. General Scheme for Constructing Generalized Traveling-Wave Solutions by the Splitting Method for Evolution Equations

  4. Example 1. Nonlinear Heat Equation Consider the nonlinear heat equation We look for generalized traveling-wave solutions of the form The functions w(z), j(t), y(t), and f (w) are to be determined. Substitute (2) into (1) and divide by w'z to obtain On expressing x from (2) in terms of z and substituting into (3), we get a functional-differential equation in two variables, t and z,

  5. Example 1. Nonlinear Heat Equation (continued) The functional differential equation can be rewritten as the standard bilinear functional equation with Substituting these expressions into the solution of the 4-term functional equation (Lecture 1) yields the determining system of ordinary differential equations where A1, A2, A3, A4 are arbitrary constants.

  6. Example 1. Nonlinear Heat Equation (continued) Determining system of ordinary differential equations: The solution to the determining system of ODEs is given by where C1, …, C4 are arbitrary parameters, A4 0. The dependence f = f (w) is defined by the last two relations in parametric form (z is treated as the parameter).

  7. Example 2. Nonlinear Heat Equation Again consider the nonlinear heat equation Differentiating (3) with respect to z yields where We now look for functional separable solutions of the special form Substitute (2) into (1) and divide by w'z to obtain Expressions (5) should then be substituted into the solution of the functional equation (4) to obtain the determining system of ODEs (see Lecture 1). where

  8. Example 3. Mass and Heat Transfer with Volume Reaction Nonlinear equation: First functional separable solution Let the function f = f (w) be arbitrary and let g = g(w) be defined by In this case, there is a functional separable solution defined implicitly by where C1, C2 are arbitrary numbers.

  9. Mass and Heat Transfer with Volume Reaction (continued) Second functional separable solution Let now g = g(w) be arbitrary and let f = f (w) be defined by Nonlinear equation: where A1, A2, A3 are some numbers. Then there are generalized traveling-wave solutions of the form where w(Z) is determined by inverting the second relation in (*) and C1, C2 are arbitrary numbers.

  10. Mass and Heat Transfer with Volume Reaction (continued) Third functional separable solution Let now g = g(w) be arbitrary and let f = f (w) be defined by Nonlinear equation: where A4  0. Then there are generalized traveling-wave solutions of the form where the function w(Z) is determined by the inversion of relation (*) and C1, C2 are arbitrary numbers.

  11. Mass and Heat Transfer with Volume Reaction (continued) Fourth functional separable solution Let the functions f = f (w) and g = g(w) be defined as follows: Nonlinear equation: where j(w) is an arbitrary function and a, b, c are any numbers; the prime denotes the derivative with respect to w. Then there are functional separable solutions defined implicitly by

  12. Mass and Heat Transfer with Volume Reaction (continued) Fifth functional separable solution Let the functions f = f (w) and g = g(w) be defined as follows: Nonlinear equation: where j(w) is an arbitrary function and a is any numbers; the prime denotes the derivative with respect to w. Then there are functional separable solutions defined implicitly by where C1, C2 are arbitrary numbers.

  13. Mass and Heat Transfer with Volume Reaction (continued) Sixth functional separable solution Let the functions f = f (w) and g = g(w) be defined as follows: Nonlinear equation: where V(z) is an arbitrary function of z; A, B are arbitrary constants (AB  0); and the function z = z(w) is determined implicitly by with C1 being an arbitrary constant. Then there is a functional separable solution of the form (*) where with C2, C3 being arbitrary constants.

  14. Nonlinear Schrödinger Equation with Cubic Nonlinearity Equation: Exact solution: where the functions a=a(t), b=b(t), a=a (t), b=b (t), g=g (t)are determined by the system of ODEs

  15. Nonlinear Schrödinger Equation of General Form Equation: 1. Exact solution: where A, B, C are arbitrary real constants, and the function j=j (t)is determined by the ordinary differential equation 2. Exact solutions: where C1, C2, C3 are arbitrary real constants.

  16. Reference A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2003

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